Lesson Vocabulary 14-2 Reﬂ...

LessonRefl ections

Chapter 14

14-2Vocabulary

BIG IDEA

Vocabularyperpendicular

midpoint

bisector

perpendicular bisector

re� ection image of A over m

re� ecting line, line of re� ection

BIG IDEA BIG IDEA The re� ection image of a � gure over a line can be found by folding the � gure over the line, or by using the idea of perpendicular bisector.

Even if you have never studied refl ections before, you already have experience with them. When you look in a mirror, you see refl ection images of objects that look like the original. Other surfaces, such as puddles of water and sheets of metal, also produce refl ections. This is the idea behind refl ections in the plane.

MATERIALS thin paper such as patty paper, a pencil, and a protractor

Step 1 Trace the trapezoid STAR and the line m on a piece of thin paper.

Step 2 Imagine re� ecting the image across the line. (You may have previously used the term “� ip” to describe re� ections. The term “� ip” has other meanings besides re� ection, so we use the term “re� ect.”) Predict and draw where you think the re� ection image would be.

SMP08TM2_SE_C06_T_0221

S

R A

T m

Step 3 Fold the thin paper along the line and trace the actual re� ection image. Call the image S′T′A′R′.

Step 4 Compare your predicted image with S′T′A′R′. Write about how your predicted image compared to the actual image.

Step 5 Draw __SS′ ,

_TT′ ,

__AA′ , and

__RR′ . What do you notice about the

four segments?

ActivityActivity

14-2 Refl ections

9 Some Important Geometry Ideas

Step 6 Measure the angle __

SS′ makes with m. What appears to be true about

_ TT′ and m,

__ AA′ and m, and

__ RR′ and m?

Step 7 Make T * the intersection of m and __

T T′ . Measure _

TT′ and __

T *T′ . What appears to be true? Does this seem to be true for

__ SS′ , __

AA′ and __

RR′ ?

When lines form right angles, we say they are perpendicular. In the Activity, m and SS′ are perpendicular, written m ⊥

__ SS′ .

QY1

The Defi nition of Refl ection ImageHere is a precise way to locate the refl ection image of a point over a line.

Begin with line m and point E as shown below.


m m

n

E

M

mE

n

E

M

E'

Step 1Given Step 2

Step 1 Draw a dashed line n perpendicular to line m containing E. Call the point of intersection M.

Step 2 Find EM. Draw the image E′ of point E on the other side of line m, such that EM = E′M.

Look at the fi gure in Step 2. The point M is on −−−

EE′ and is the same distance from the endpoints of the segment. It is exactly in the middle of the segment. For this reason, it is called the midpoint of the segment. Because line m cuts

−−− EE′ at its midpoint, line m is called a bisector

of −−−

EE′ . And because m is also perpendicular to −−−

EE′ , m is called the perpendicular bisector of

−−− EE′ .

QY2

QY1

Use your results from the Activity to write 3 other perpendicular statements.

QY1


QY1


QY2

Suppose C is on line m. Where will its re� ection image over line m lie?

QY2

Suppose C is C is C on line m.Where will its re� ection image over line m lie?

QY2

Suppose C is on line m. Where will its re� ection image over line m lie?

Re� ections 10

Lesson 14-2

Chapter 14

From these characteristics, we can precisely defi ne the refl ection image of a point over a line.

Refl ection Image of A over m

Let A be a point and m be a line. If A is not on m, then the reflection image of A over m is the point A′ so that m is the perpendicular bisector of

−− AA′ . If A is on m, then A coincides with

its re� ection image.

The line m is called the reflecting line, or the line of reflection.

Refl ecting a Figure over a LineTo refl ect a more complicated fi gure, fi nd the images of important points on the fi gure. It is enough to refl ect each vertex of a polygon and then draw the image. Here are �WHO and its refl ection image over line z.


H z

W

O

H�

W�O�

Line z is the perpendicular bisector of each segment that connects a point to its refl ection image.

Figures that intersect the refl ecting line can still be refl ected over that line. When a point is on the refl ecting line, it coincides with its image, so points B and C on refl ecting line m at the right are on both the fi gure JADE and its refl ection image J′A′D′E′.

A fi gure and its refl ection image are congruent, so �WHO � �W′H′O′. Quadrilaterals JADE and J′A′D′E′ are also congruent. As a result, corresponding sides and angles of these fi gures have the same measure.

QY3


C

B

E�D�

J� J

m

A�A

DE


C

B

E�D�

J� J

m

A�A

DE

QY3

Name the sides and the angles in JADE and J′A′D′E′ that are congruent.

QY3

Name the sides and the angles in JADE and JADE and JADE J′A′A′ ′D′E′that are congruent.

QY3

Name the sides and the angles in JADE and J′A′D′E′ that are congruent.

Chapter 14


QuestionsCOVERING THE IDEAS

In 1 and 2, use the diagram at the right. Point H′ is the re� ection image of point H over line �.

1. Fill in the Blank a. � is called the ? . b. H is called the ? . c. m∠BEH = ? . d.

−−− HH′ is ? to �.

e. H and H′ are the same ? from �. f. The refl ection image of B over � is ? .

2. What relationship exists between __

HH' and �? Express the relationship symbolically.

3. True or False A fi gure and its refl ection image are congruent.

In 4–6, refer to �DEF at the right.

4. If �DEF is refl ected over � �� DF , explain why D coincides with its image.

5. If �DEF is refl ected over � �� DF , and E' is the image of E, explain why DEFE' is a rhombus.

6. In Question 5, explain why ∠E � ∠DE'F.

7. Trace the drawing at the right. Then draw the refl ection images of Q, R, and S over line n.

In 8 and 9, trace the drawing. Then draw the re� ection image of the given � gure over the given line and name the congruent sides and angles.

8.


DC j

E

9.


H

Gn

IF


HH�

D

B

E

�


HH�

D

B

E

�


D F

E


D F

E


Q

S

n

R


Q

S

n

R

Lesson 14-2

Re� ections 12

Chapter 14

APPLYING THE MATHEMATICS

In 10 and 11, re� ect the � gure over the re� ecting line.

10.


11.


12. Suppose the point A = (1, 3) is refl ected over the x-axis. a. What are the coordinates of its image A′?

(Hint: Draw a picture.) b. What are the coordinates of A∗, the refl ection image of A′

over the y -axis? c. If the point (a, b) is refl ected over the x-axis, what are the

coordinates of the image point? d. If the point (a, b) is refl ected over the y -axis, what are the

coordinates of the image point?

QY ANSWERS

1. m ⊥ __

T T′ ; m ⊥ __

AA′ ;

m ⊥ __

RR′

2. It will lie on itself.

3. −− JA �

−− J ′A′ , −−

AD � −−−

A′D′ , −−

DE � −−−

D′E′ , −− E J �

−− E′J ′ ,

∠J � ∠J ′, ∠A � ∠A′, ∠D � ∠D′, ∠E � ∠E′.


Lesson Vocabulary 14-2 Reﬂ...

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